To make this relationship between (inter)stripes and dimension-1 topological features more precise [1]}, it is useful to discuss birth and death times [2]}. The “birth time" of a topological feature is the value \(\varepsilon = \varepsilon _\text{birth}\)  at which the feature first appears, and the “death time" is the value \(\varepsilon = \varepsilon _\text{death}\)  at which the feature disappears [1]}. A feature's “persistence" is \(\varepsilon _\text{death}-\varepsilon _\text{birth}\) , and this information is often visualized using barcodes and persistence diagrams. We show the barcode and persistence diagram associated with dimension-1 topological features for an example of three interstripes in Figure REF .
In a barcode, we draw a horizontal bar starting at \(\varepsilon _\text{birth}\)  and ending at \(\varepsilon _\text{death}\)  for each topological feature [2]}; these bars are stacked vertically. A persistence diagram is a scatter plot with birth time as the \(x\) -axis and death time as the \(y\) -axis; each point \((\varepsilon _\text{birth},\varepsilon _\text{death})\)  refers to the birth and death times of a feature. As we show in Figure REF (c), there are three long bars, corresponding to three highly persistent loops. Specifically, McGuirl et al. [1]} defined the number of interstripes in ABM [6]} patterns by counting the number of \(X^\text{d}\)  loops with persistence greater than a persistence threshold, and birth time less than a birth-time cutoff. The persistence threshold and birth-time cutoff are hyper-parameters. In §REF , we build on the work [1]} to make this approach more flexible for on-lattice stripe patterns and reliably choose hyper-parameter values.
